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A108756
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A triangle related to the Jacobsthal polynomials.
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3
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1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 4, 5, 1, 1, 3, 6, 6, 7, 1, 1, 1, 10, 15, 8, 9, 1, 1, 4, 10, 21, 28, 10, 11, 1, 1, 1, 20, 35, 36, 45, 12, 13, 1, 1, 5, 15, 56, 84, 55, 66, 14, 15, 1, 1, 1, 35, 70, 120, 165, 78, 91, 16, 17, 1, 1, 6, 21, 126, 210, 220, 286, 105, 120, 18, 19, 1, 1
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OFFSET
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0,7
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COMMENTS
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Riordan array ((1 + x - x^2)/(1 - x^2)^2, x/(1 - x^2)^2). Row sums are A108742. Diagonal sums are Fibonacci(n+1) = A000045(n+1). Corresponding diagonals triangle is A102426.
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LINKS
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FORMULA
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Number triangle: T(n, k) = binomial(floor((n + k + 1)/2) + k, floor((n + k)/2 - k)) for 0 <= k <= n.
Bivariate g.f.: Sum_{n, k > = 0} T(n,k) * x^n * y^k = (1 + x - x^2)/((1 - x^2)^2 - x * y). (Here, we assume T(n, k) = 0 for n < k. Because T(n, k) = A102426(n + 1 + k, k), we may use Tom Copeland's g.f. of the latter array, to get the g.f. of the current triangular array.)
G.f. for column k >= 0: (1 + x - x^2) * x^k/(1 - x^2)^(2*k + 2).
Recurrence: T(n, k) = T(n - 2, k) + T(n - (1 - (-1)^(n + k))/2, k - (1 + (-1)^(n + k))/2), for n >= 3 and 1 <= k <= n - 2, starting with T(n, n) = 1 = T(n + 1, n) for n >= 0, T(n, 0) = 1 when n is even >= 0, and T(n, 0) = (n + 1)/2 when n is odd >= 1.
Another recurrence: T(n, k) = T(n - 1, k - 1) + 2*T(n - 2, k) - T(n - 4, k) for n >= 4 and 1 <= k <= n - 4. (This follows from the fact that the denominator of the bivariate g.f. is x^0 * y^0 - x^1 * y^1 - 2 * x^2 * y^0 - x^4 * y^0.)
(End)
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EXAMPLE
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Triangle begins (with rows n >= 0 and columns k >= 0) as follows:
1;
1, 1;
1, 1, 1;
2, 3, 1, 1;
1, 4, 5, 1, 1;
3, 6, 6, 7, 1, 1;
1, 10, 15, 8, 9, 1, 1;
4, 10, 21, 28, 10, 11, 1, 1;
1, 20, 35, 36, 45, 12, 13, 1, 1;
5, 15, 56, 84, 55, 66, 14, 15, 1, 1; ...
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MATHEMATICA
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Table[Binomial[Floor[(n+k+1)/2]+k, Floor[(n+k)/2]-k], {n, 0, 12}, {k, 0, n} ]//Flatten (* G. C. Greubel, May 29 2019 *)
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PROG
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(PARI) {T(n, k) = binomial(floor((n+k+1)/2)+k, floor((n+k)/2)-k)}; \\ G. C. Greubel, May 29 2019
(Magma) [[Binomial(Floor((n+k+1)/2)+k, Floor((n+k)/2)-k): k in [0..n]]: n in [0..12]]; // G. C. Greubel, May 29 2019
(Sage) [[binomial(floor((n+k+1)/2)+k, floor((n+k)/2)-k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, May 29 2019
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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